What is the ruling on using products that contain glycerin, if its source is unknown (plant-based or animal-based)?

There is no objection to using glycerin of unknown origin, for the default is permissibility until proven otherwise. Glycerin is mostly prepared in laboratories from molasses, and it may be prepared from soap. Rarely is it derived from pig fat. The default for everything Allah has created for us on earth is permissibility; whoever claims something is forbidden must provide proof.